problem solving multiple trigonometric equations

On Tue, Jun 22, 2010 at 04:38:17PM -0500, Barton Willis wrote:
> -----maxima-bounces at math.utexas.edu wrote: -----
> 
> >Very recently I started using Maxima and already got some usefull results.
> >However, I now stumbled on a problem with solving mutliple equations
> >involving
> >trigonometric functions. A simple (not very usefull) example:
> >
> >(%i1) solve([cos(x) = 1/2, sin(y) = 1/2], [x,y]);
> >(%o1)                                 []
> 
> Maxima has two alternative solve functions. I know little about one
> of these, but the optional package to_poly_solver can solve trigonometric 
> equations that can be converted into polynomial equations that algsys can solve.
> Try this:
> 
> (%i1) load(to_poly_solver)$
> 
> (%i8) %solve([cos(x) = 1/2, sin(y) = 1/2], [x,y], 'simpfuncs = ['ratexpand]);
> 
> (%o8) %union([x=2*%pi*%z58-%pi/3,y=2*%pi*%z58+(5*%pi)/6],[x=2*%pi*%z60-%pi/3,y=2*%pi*%z60+%pi/6],[x=2*%pi*%z62+%pi/3,y=2*%pi*%z62+(5*%pi)/6],[x=2*%pi*%z64+%pi/3,y=2*%pi*%z64+%pi/6])
> 
> The user documentation for to_poly_solve is 
> 
>    .../maxima/share/contrib/topoly-user-doc.html
> 
> where ... depends on the location of your Maxima. Welcome to Maxima.
> Let us know if you have questions about to_poly_solver or Maxima in
> general.

Thanks Barton, for the quick reply.
I compiled maxima-5.21.1, which allowed me to load the to_poly_solver
package:

(%i1) load("to_poly_solver");
  Loading maxima-grobner $Revision: 1.6 $ $Date: 2009/06/02 07:49:49 $
  define: warning: redefining the built-in function prog1
  define: warning: redefining the built-in function symbolcheck
  define: warning: redefining the built-in function push
  define: warning: redefining the built-in function pop
  define: warning: redefining the built-in function tr_ev
  WARNING: DEFUN/DEFMACRO: redefining macro OPAPPLY in
           /home/obooij/src/maxima-5.21.1/share/contrib/fourier_elim/fourier_elim.
  lisp,
           was defined in
           /home/obooij/src/maxima-5.21.1/share/contrib/topoly.lisp
  WARNING: DEFUN/DEFMACRO: redefining macro OPCONS in
           /home/obooij/src/maxima-5.21.1/share/contrib/fourier_elim/fourier_elim.
  lisp,
           was defined in
           /home/obooij/src/maxima-5.21.1/src/binary-clisp/nummod.fas
  WARNING: DEFUN/DEFMACRO: redefining function SIMP-ISREAL-P in
           /home/obooij/src/maxima-5.21.1/share/contrib/to_poly_solve_extra.lisp,
           was defined in
           /home/obooij/src/maxima-5.21.1/share/contrib/topoly.lisp
  (%o1)   /home/obooij/src/maxima-5.21.1/share/contrib/to_poly_solver.mac


I hope I can safely ignore these warnings.
This indeed allowed me to solve the equation:

  (%i2) %solve([cos(x) = 1/2, sin(y) = 1/2], [x,y], 'simpfuncs = ['ratexpand]);
                                 %pi                   5 %pi
  (%o2) %union([x = 2 %pi %z15 - ---, y = 2 %pi %z15 + -----], 
                                  3                      6
                    %pi                   %pi
  [x = 2 %pi %z17 - ---, y = 2 %pi %z17 + ---], 
                     3                     6
                    %pi                   5 %pi
  [x = 2 %pi %z19 + ---, y = 2 %pi %z19 + -----], 
                     3                      6
                    %pi                   %pi
  [x = 2 %pi %z21 + ---, y = 2 %pi %z21 + ---])
                     3                     6

However, a somewhat more involved (although still simple) example does not seem to work:

%solve([cos(x) = 1/2, cos(x)*sin(y) = 1/3], [x,y], 'simpfuncs = ['ratexpand]);
Unable to solve
Unable to solve
Unable to solve
                       1                  1
(%o3) %solve([cos(x) = -, cos(x) sin(y) = -], [x, y], simpfuncs = [ratexpand])
                       2                  3
 
I didn't have a look at topoly-user-doc.html yet, so perhaps the answer is
in there (and if so: sorry for this post).

Olaf